$E(t)$ models the daily amount of energy (in kilojoules, $\text{kJ}$ ) that a solar panel in Pago Pago generates, $t$ days after the autumn equinox. Here, $t$ is entered in radians. $E(t) = {624}\sin\left({\dfrac{2\pi}{365}}t\right) + {8736}$ What is the first day after the autumn equinox that the solar panel generates $8400\text{ kJ}$ ? Round your final answer to the nearest whole day.
Solution: Converting the problem into mathematical terms $E(t) = {624}\sin\left({{\dfrac{2\pi}{365}}}t\right) + {8736}$ has a period of $\dfrac{2\pi}{{\scriptsize\dfrac{2\pi}{365}}}=365$ days. We want to find the first solution to the equation $E(t)=8400$ within the period $0<t<365$. The answer The equation's two solutions within the desired period (rounded to the nearest whole day) are $216$ and $332$. Therefore, the first time that the panel generates $8400\text{ kJ}$ is $216$ days after the autumn equinox.